In this paper we investigate the number of rooted loopless unicursal planar maps and present some formulae for such maps with up to three parameters: the number of edges and the valencies of the two odd vertices.
Citation
Shude Long, Junliang Cai. Enumeration of Rooted Loopless Unicursal Planar Maps[J], Ars Combinatoria, Volume 117. 131-146. .